One-dimensional Kirchhoff equation (Q5952710)

The paper deals with the Cauchy problem of the form NEWLINE\[NEWLINE\begin{gathered} u_{tt} (x,t)-\gamma \left(\int^\infty_{-\infty} u^2_x(x,t)dx \right)u_{xx}(x,t) =0,\\ u(x,0)= \varphi(x),\;u_t(x,0) =\psi(x),\;x\in \mathbb{R},\end{gathered}NEWLINE\]NEWLINE where \(\gamma: [0,\infty)\to [0,\infty)\) is a given Lipschitz \(C^1\) function, \(\varphi \in C^3(\mathbb{R})\), \(\psi\in C^2(\mathbb{R})\) are ``small'' in suitable norm and \(\varphi', \varphi'',\varphi''', \psi',\psi''\) are ``well behaved'' at infinity. The author proves the existence of a global solution.